Sen. Lindsey Graham , R-S.C., said the deal would establish a pathway to citizenship for the nearly 2 million undocumented immigrants brought to the U.S. as children and protected under the Deferred Action for Childhood Arrivals program, or DACA, Bloomberg reported. Graham also said the deal will address border security.

Graham said the deal would not, however, deal with the so-called diversity visa lottery, which prioritizes immigration from countries that send fewer immigrants to the U.S., or family reunification, which immigration hardliners have dubbed "chain migration."

Axios reported on Tuesday that President Donald Trump will veto an immigration bill that doesn't include the reforms he wants. The report cited a senior administration official. Trump has pushed to end the family reunification and visa lottery systems.

Trump decided in September to end the Obama-era DACA but gave Congress until March to fix it.

The offices of Sens. Jeff Flake , Susan Collins , Graham and Kaine did not immediately respond to CNBC's requests for comment.